Finite Group Databasebeta

← C83×#9377 #35237201

Subgroups — C83×#9377 #35237201

Order 21248.

434 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C83×#3239C211yesmaximal, characteristic
2C83×#2992C211yesmaximal, characteristic
3C83×#1981C211yesmaximal, characteristic, complemented
4C83×#1556C211yesmaximal, characteristic, complemented
5C83×#1509C211yesmaximal, characteristic
6C83×#1290C211yesmaximal, characteristic, complemented
7C83×#1285C211yesmaximal, characteristic, complemented
8C2×C4⋊Q8×C83C2211yescharacteristic
9#529×C83C2211yescharacteristic
10#501×C83C412yescomplemented
11#489×C83C412yescomplemented
12#484×C83—21no—
13#481×C83—21no—
14#475×C83—21no—
15#474×C83—21no—
16#446×C83—21no—
17#435×C83—21no—
18#435×C83—21no—
19C8×D4×C83—21no—
20#433×C83—21no—
21#432×C83C2211yescharacteristic
22#432×C83C2211yescharacteristic
23#432×C83C2211yescharacteristic
24#424×C83—21no—
25C2×C4⋊C8×C83C2211yescharacteristic
26C2×#103×C83—21no—
27C2×#103×C83—21no—
28C4×C4⋊C4×C83C2211yescharacteristic
29#332×C83—21no—
30#332×C83—21no—
31#328×C83—21no—
32#327×C83—21no—
33C22×Q8×C83—21no—
34C2×M16×C83—21no—
35C22×C8×C83—21no—
36C22×C8×C83D411yescharacteristic
37C22×C8×C83D411yescharacteristic
38C4⋊Q8×C83—21no—
39C4⋊Q8×C83—21no—
40C4⋊Q8×C83—21no—
41C4⋊Q8×C83C2×C412yes—
42#126×C83—21no—
43#125×C83C2×C411yescharacteristic
44#124×C83C2×C411yescharacteristic
45#123×C83—21no—
46C22⋊Q8×C83D411yescharacteristic
47#121×C83D411yescharacteristic
48C4×Q8×C83D411yescharacteristic
49C4×D4×C83—21no—
50C4×D4×C83D411yescharacteristic
51#117×C83—21no—
52C2×C4⋊C4×C83—21no—
53C2×C4⋊C4×C83—21no—
54C2×C4⋊C4×C83—21no—
55C2×C42×C83—21no—
56C2×C42×C83C2311yescharacteristic
57#107×C83—22no—
58#106×C83—22no—
59C4⋊C8×C83—21no—
60C4⋊C8×C83—21no—
61C4⋊C8×C83—21no—
62C4⋊C8×C83—21no—
63C4⋊C8×C83—21no—
64C4⋊C8×C83—21no—
65C4⋊C8×C83D411yescharacteristic
66C4⋊C8×C83D411yescharacteristic
67#103×C83—22no—
68#103×C83—42no—
69#103×C83—42no—
70#102×C83—22no—
71C22⋊C8×C83—21no—
72C22⋊C8×C83—21no—
73C22⋊C8×C83—21no—
74C22⋊C8×C83—21no—
75C22⋊C8×C83—21no—
76#97×C83—21no—
77#97×C83—21no—
78#97×C83—21no—
79#97×C83C2×C412yes—
80C4×C8×C83—21no—
81#95×C83—21no—
82C2×Q8×C83—22no—
83C2×Q8×C83—41no—
84C2×Q8×C83—41no—
85C2×Q8×C83—41no—
86C2×Q8×C83—41no—
87C2×Q8×C83—41no—
88C2×Q8×C83C22⋊C411yescharacteristic
89C2×D4×C83—21no—
90C2×D4×C83C22⋊C411yescharacteristic
91C22×C4×C83—21no—
92C22×C4×C83—41no—
93C22×C4×C83—41no—
94C22×C4×C83C2×D411yescharacteristic
95C22×C4×C83C2×D411yescharacteristic
96C22×C4×C83#4111yescharacteristic
97M16×C83—81no—
98C2×C8×C83—22no—
99C2×C8×C83—21no—
100C2×C8×C83—21no—
101C2×C8×C83—21no—
102C2×C8×C83—21no—
103C2×C8×C83—41no—
104C2×C8×C83—41no—
105C2×C8×C83—41no—
106C2×C8×C83—81no—
107C4⋊C4×C83—21no—
108C4⋊C4×C83—21no—
109C4⋊C4×C83—21no—
110C4⋊C4×C83—21no—
111C4⋊C4×C83—22no—
112C4⋊C4×C83—22no—
113C4⋊C4×C83—22no—
114C4⋊C4×C83—41no—
115C4⋊C4×C83—41no—
116C4⋊C4×C83—41no—
117C4⋊C4×C83C22⋊C411yescharacteristic
118C4⋊C4×C83C22⋊C411yescharacteristic
119C22⋊C4×C83—21no—
120C22⋊C4×C83—21no—
121C22⋊C4×C83—21no—
122C22⋊C4×C83—21no—
123C22⋊C4×C83—21no—
124C42×C83—21no—
125C42×C83—81no—
126C42×C83C22×C411yescharacteristic
127C42×C83C2×D411yescharacteristic
128C42×C83C2×D411yescharacteristic
129C42×C83#4111yescharacteristic
130C23×C83—21no—
131C23×C83#12111yescharacteristic
132Q8×C83—41no—
133Q8×C83—42no—
134Q8×C83—42no—
135Q8×C83—81no—
136Q8×C83—81no—
137D4×C83—41no—
138D4×C83—41no—
139D4×C83—41no—
140C2×C4×C83—21no—
141C2×C4×C83—21no—
142C2×C4×C83—21no—
143C2×C4×C83—21no—
144C2×C4×C83—21no—
145C2×C4×C83—21no—
146C2×C4×C83—21no—
147C2×C4×C83—41no—
148C2×C4×C83—41no—
149C2×C4×C83—41no—
150C2×C4×C83—42no—
151C2×C4×C83—42no—
152C2×C4×C83—81no—
153C2×C4×C83—81no—
154C2×C4×C83—81no—
155C2×C4×C83—81no—
156C2×C4×C83C2×C22⋊C411yescharacteristic
157C2×C4×C83C4×D411yescharacteristic
158C2×C4×C83C4×D411yescharacteristic
159C2×C4×C83#12011yescharacteristic
160C2×C4×C83#12111yescharacteristic
161C2×C4×C83#12311yescharacteristic
162C8×C83—42no—
163C8×C83—41no—
164C8×C83—41no—
165C8×C83—41no—
166C8×C83—41no—
167C8×C83—81no—
168C22×C83—21no—
169C22×C83—21no—
170C22×C83—41no—
171C22×C83—41no—
172C22×C83—41no—
173C22×C83#38611yescharacteristic
174C22×C83#47111yescharacteristic
175C22×C83#47311yescharacteristic
176C4×C83—21no—
177C4×C83—21no—
178C4×C83—21no—
179C4×C83—21no—
180C4×C83—21no—
181C4×C83—41no—
182C4×C83—41no—
183C4×C83—41no—
184C4×C83—82no—
185C4×C83—82no—
186C4×C83#45311yescharacteristic
187C4×C83#45411yescharacteristic
188#9377C8311yesmaximal, characteristic, direct product factor, Sylow
189C2×C83—21no—
190C2×C83—21no—
191C2×C83—81no—
192C2×C83#188311yescharacteristic
193C2×C83#190811yescharacteristic
194C2×C83#190911yescharacteristic
195#3239C2×C8311yescharacteristic
196#2992C2×C8311yescharacteristic
197#1981C2×C8311yescharacteristic, complemented
198#1556C2×C8311yescharacteristic, complemented
199#1509C2×C8311yescharacteristic
200#1290C2×C8311yescharacteristic, complemented
201#1285C2×C8311yescharacteristic, complemented
202C83#937711yescharacteristic, direct product factor, Sylow
203C2×C4⋊Q8C22×C8311yescharacteristic
204#529C22×C8311yescharacteristic
205#501C4×C8312yescomplemented
206#489C4×C8312yescomplemented
207#484—21no—
208#481—21no—
209#475—21no—
210#474—21no—
211#446—21no—
212#435—21no—
213#435—21no—
214C8×D4—21no—
215#433—21no—
216#432C22×C8311yescharacteristic
217#432C22×C8311yescharacteristic
218#432C22×C8311yescharacteristic
219#424—21no—
220C2×C4⋊C8C22×C8311yescharacteristic
221C2×#103—21no—
222C2×#103—21no—
223C4×C4⋊C4C22×C8311yescharacteristic
224#332—21no—
225#332—21no—
226#328—21no—
227#327—21no—
228C22×Q8—21no—
229C2×M16—21no—
230C22×C8—21no—
231C22×C8D4×C8311yescharacteristic
232C22×C8D4×C8311yescharacteristic
233C4⋊Q8—21no—
234C4⋊Q8—21no—
235C4⋊Q8—21no—
236C4⋊Q8C2×C4×C8312yes—
237#126—21no—
238#125C2×C4×C8311yescharacteristic
239#124C2×C4×C8311yescharacteristic
240#123—21no—
241C22⋊Q8D4×C8311yescharacteristic
242#121D4×C8311yescharacteristic
243C4×Q8D4×C8311yescharacteristic
244C4×D4—21no—
245C4×D4D4×C8311yescharacteristic
246#117—21no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—21no—
250C2×C42—21no—
251C2×C42C23×C8311yescharacteristic
252#107—22no—
253#106—22no—
254C4⋊C8—21no—
255C4⋊C8—21no—
256C4⋊C8—21no—
257C4⋊C8—21no—
258C4⋊C8—21no—
259C4⋊C8—21no—
260C4⋊C8D4×C8311yescharacteristic
261C4⋊C8D4×C8311yescharacteristic
262#103—22no—
263#103—42no—
264#103—42no—
265#102—22no—
266C22⋊C8—21no—
267C22⋊C8—21no—
268C22⋊C8—21no—
269C22⋊C8—21no—
270C22⋊C8—21no—
271#97—21no—
272#97—21no—
273#97—21no—
274#97C2×C4×C8312yes—
275C4×C8—21no—
276#95—21no—
277C2×Q8—22no—
278C2×Q8—41no—
279C2×Q8—41no—
280C2×Q8—41no—
281C2×Q8—41no—
282C2×Q8—41no—
283C2×Q8C22⋊C4×C8311yescharacteristic
284C2×D4—21no—
285C2×D4C22⋊C4×C8311yescharacteristic
286C22×C4—21no—
287C22×C4—41no—
288C22×C4—41no—
289C22×C4C2×D4×C8311yescharacteristic
290C22×C4C2×D4×C8311yescharacteristic
291C22×C4#41×C8311yescharacteristic
292M16—81no—
293C2×C8—22no—
294C2×C8—21no—
295C2×C8—21no—
296C2×C8—21no—
297C2×C8—21no—
298C2×C8—41no—
299C2×C8—41no—
300C2×C8—41no—
301C2×C8—81no—
302C4⋊C4—21no—
303C4⋊C4—21no—
304C4⋊C4—21no—
305C4⋊C4—21no—
306C4⋊C4—22no—
307C4⋊C4—22no—
308C4⋊C4—22no—
309C4⋊C4—41no—
310C4⋊C4—41no—
311C4⋊C4—41no—
312C4⋊C4C22⋊C4×C8311yescharacteristic
313C4⋊C4C22⋊C4×C8311yescharacteristic
314C22⋊C4—21no—
315C22⋊C4—21no—
316C22⋊C4—21no—
317C22⋊C4—21no—
318C22⋊C4—21no—
319C42—21no—
320C42—81no—
321C42C22×C4×C8311yescharacteristic
322C42C2×D4×C8311yescharacteristic
323C42C2×D4×C8311yescharacteristic
324C42#41×C8311yescharacteristic
325C23—21no—
326C23#121×C8311yescharacteristic
327Q8—41no—
328Q8—42no—
329Q8—42no—
330Q8—81no—
331Q8—81no—
332D4—41no—
333D4—41no—
334D4—41no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—42no—
346C2×C4—42no—
347C2×C4—81no—
348C2×C4—81no—
349C2×C4—81no—
350C2×C4—81no—
351C2×C4C2×C22⋊C4×C8311yescharacteristic
352C2×C4C4×D4×C8311yescharacteristic
353C2×C4C4×D4×C8311yescharacteristic
354C2×C4#120×C8311yescharacteristic
355C2×C4#121×C8311yescharacteristic
356C2×C4#123×C8311yescharacteristic
357C8—42no—
358C8—41no—
359C8—41no—
360C8—41no—
361C8—41no—
362C8—81no—
363C22—21no—
364C22—21no—
365C22—41no—
366C22—41no—
367C22—41no—
368C22#386×C8311yescharacteristic
369C22#471×C8311yescharacteristic
370C22#473×C8311yescharacteristic
371C4—21no—
372C4—21no—
373C4—21no—
374C4—21no—
375C4—21no—
376C4—41no—
377C4—41no—
378C4—41no—
379C4—82no—
380C4—82no—
381C4#453×C8311yescharacteristic
382C4#454×C8311yescharacteristic
383C2—21no—
384C2—21no—
385C2—81no—
386C2C83×#188311yescharacteristic
387C2C83×#190811yescharacteristic
388C2C83×#190911yescharacteristic
subgroupG/H
center, socleC22×C83#386
commutatorC42C22×C4×C83
Frattini, cosocleC2×C42C23×C83
prime radical#121×C83D4
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C83 ◁ C2×C42×C83 ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
3. C83×#1981C2semidirect
4. C83×#1556C2semidirect
6. C83×#1290C2semidirect
7. C83×#1285C2semidirect
10. #501×C83C4semidirect
11. #489×C83C4semidirect
188. #9377C83direct (N × H)
197. #1981C2×C83semidirect
198. #1556C2×C83semidirect
200. #1290C2×C83semidirect
201. #1285C2×C83semidirect
202. C83#9377direct (N × H)
205. #501C4×C83semidirect
206. #489C4×C83semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C83×#32392
C83×#29922
C83×#19812
C83×#15562
C83×#15092
C83×#12902
C83×#12852
#937783